Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(5,6)(7,12)(8,11)(9,10);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,20)(16,17)(18,19);(6,10)(7,11)(14,19)(15,16);(13,15,18,16)(14,20,19,17);(5,7,9,11)(6,8,10,12)(13,16,18,15)(14,17,19,20);(5,8)(6,11)(7,10)(9,12)(13,17)(14,15)(16,19)(18,20)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72233751, #72233767, #72234448, … |
| maximal normal subgroup of | #72233751, #72233767, #72234448, … |
| maximal quotient of | #72256742, #72262198, #72263302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7644608);